Conservation of momentum on a zero-friction floor

If an object is moving with velocity $v$ and you slice it in half perpendicular to the plane of the floor, each piece moves with velocity $v$ and thus horizontal momentum is conserved since the force is perpendicular to the floor. If there’s a spontaneous breakup such as an explosion, then the final momenta of the pieces depends on the force that each piece exerts on the other. If there’s a third object moving in a horizontal plane that hits the object, splitting it in half, then the total horizontal momentum of the pieces will definitely not be $2v$, since now there is an external force on the system in consideration (I.e. the original object that breaks into two).

TL; DR: How you break the object in half matters.